Mathematical Analysis of an SIQR Influenza Model with Imperfect Quarantine

The identification of mechanisms responsible for recurrent epidemic outbreaks, such as age structure, cross-immunity and variable delays in the infective classes, has challenged and fascinated epidemiologists and mathematicians alike. This paper addresses, motivated by mathematical work on influenza models, the impact of imperfect quarantine on the dynamics of SIR-type models. A susceptible-infectious-quarantine-recovered (SIQR) model is formulated with quarantined individuals altering the transmission dynamics process through their possibly reduced ability to generate secondary cases of infection. Mathematical and numerical analyses of the model of the equilibria and their stability have been carried out. Uniform persistence of the model has been established. Numerical simulations show that the model supports Hopf bifurcation as a function of the values of the quarantine effectiveness and other parameters. The upshot of this work is somewhat surprising since it is shown that SIQR model oscillatory behavior, as shown by multiple researchers, is in fact not robust to perturbations in the quarantine regime.